Quotes by Mathematicians - Page 6

The Wine of Life keeps oozing drop by drop, The Leaves of Life keep falling one by one.

Doing mathematics should always mean finding patterns and crafting beautiful and meaningful explanations.

I went to Russia a Communist; but contact with those who have no doubts has intensified a thousandfold my own doubts, not as to Communism in itself, but as to the wisdom of holding a creed so firmly that for its sake men are willing to inflict widespread misery.

Those honor nature well, who teach that she can speak on everything.

I think actually if you take the analogy with other areas of engineering, and increasingly of science and even mathematics, you can see people do not have to learn the vast number of formulae they used to learn. Instead, they have to learn to use the computer effectively. This frees them, I feel, to understand concepts and the foundations while they’re learning the mechanics of the application of the theory.

For all bodies are in perpetual flux like rivers, and parts are passing in and out of them continually.

When a friend introduced me to a Bach chaconne, he started by describing it by saying that it has 256 measures (256=2^8) divided into 4 sections of 64 measures (64=2^6), and I liked it even before I heard a single note.

Even fairly good students, when they have obtained the solution of the problem and written down neatly the argument, shut their books and look for something else. Doing so, they miss an important and instructive phase of the work. ... A good teacher should understand and impress on his students the view that no problem whatever is completely exhausted.

Religion is what an individual does with his solitariness.

Myself when young did eagerly frequent Doctor and Saint and heard great argument About it and about: but evermore Came out by the same door where in I went.

As I read the Qur’an and prayed the Islamic prayers, a door to my heart was unsealed and I was immersed in an overwhelming tenderness.

Never try to discourage thinking, for you are sure to succeed.

To understand this new frontier, I will have to try to master one of the most difficult and counterintuitive theories ever recorded in the annals of science: quantum physics. Listen to those who have spent their lives immersed in this world and you will have a sense of the challenge we face. After making his groundbreaking discoveries in quantum physics, Werner Heisenberg recalled, "I repeated to myself again and again the question: Can nature possibly be so absurd as it seemed to us in these atomic experiments?" Einstein declared after one discovery, "If it is correct it signifies the end of science." Schrödinger was so shocked by the implications of what he'd cooked up that he admitted, "I do not like it and I am sorry I had anything to do with it." Nevertheless, quantum physics is now one of the most powerful and well-tested pieces of science on the books. Nothing has come close to pushing it off its pedestal as one of the great scientific achievements of the last century. So there is nothing to do but to dive headfirst into this uncertain world. Feynman has some good advice for me as I embark on my quest: "I am going to tell you what nature behaves like. If you will simply admit that maybe she does behave like this, you will find her a delightful, entrancing thing. Do not keep saying to yourself, if you can possibly avoid it, 'But how can it be like that?' because you will get 'down the drain,' into a blind alley from which nobody has yet escaped. Nobody knows how it can be like that.

In a way, mathematics is the only infinite human activity. It is conceivable that humanity could eventually learn everything in physics or biology. But humanity certainly won't ever be able to find out everything in mathematics, because the subject is infinite. Numbers themselves are infinite. That's why mathematics is really my only interest.

Everyone knows that physicists are concerned with the laws of the universe and have the audacity sometimes to think they have discovered the choices God made when He created the universe in thus and such a pattern. Mathematicians are even more audacious. What they feel they discover are the laws that God Himself could not avoid having to follow.

You all know the argument from design: everything in the world is made just so that we can manage to live in the world, and if the world was ever so little different, we could not manage to live in it. That is the argument from design. It sometimes takes a rather curious form; for instance, it is argued that rabbits have white tails in order to be easy to shoot. I do not know how rabbits would view that application.

It is the heart which experiences God not the reason.

For I do not believe God means us thus to divide life into half halves - to wear a grave face on Sunday, and to think it out-of-place to even so much as mention Him on a weekday. Do you think he cares to see only kneeling figures, and to hear only tones of prayer - and that He does not also love to see the lambs leaping in the sunlight, and to hear the merry voices of the children as they roll among the hay? Surely their innocent laughter is as sweet in His ears as the grandest anthem that ever rolled up from the 'dim religious light' of some solemn cathedral?

All that alcohol does for them is to liberate the sense of sin, which reason suppresses in saner moments.

As long as men massacre animals, they will kill each other. Indeed, he who sows the seeds of murder and pain cannot reap the joy of love.

With the magnificence of eternity before us, let time, with all its fluctuations, dwindle into its own littleness.

We sail within a vast sphere ever drifting in uncertainty driven from end to end.

To fear love is to fear life and those who fear life are already three parts dead.

I entered Princeton University as a graduate student in 1959, when the Department of Mathematics was housed in the old Fine Hall. This legendary facility was marvellous in stimulating interaction among the graduate students and between the graduate students and the faculty. The faculty offered few formal courses, and essentially none of them were at the beginning graduate level. Instead the students were expected to learn the necessary background material by reading books and papers and by organising seminars among themselves. It was a stimulating environment but not an easy one for a student like me, who had come with only a spotty background. Fortunately I had an excellent group of classmates, and in retrospect I think the "Princeton method" of that period was quite effective.

When a man tells you he knows the exact truth about anything, you are safe in inferring he is an inexact man.

For many years I have been saying that I would like to write a book (or series of books) called Physics for Mathematicians. Whenever I would tell people that, they would say, “Oh good, you're going to explain quantum mechanics, or string theory, or something like that”. And I would say, “Well that would be nice, but I can't begin to do that now; first I have to learn elementary physics, so the first thing I will be writing will be Mechanics for Mathematicians”. So then people would say, “Ah, so you're going to be writing about symplectic structures”, or something of that sort. And I would have to say, “No, I'm not trying to write a book about mathematics for mathematicians, I'm trying to write a book about physics for mathematicians”; …… it's elementary mechanics that I don't understand. … I mean, for example, that I don't understand this – lever.... Most of us know the law of the lever, but this law is simply a quantitative statement of exactly how amazing the lever is, and doesn't give us a clue as to why it is true, how such a small force at one end can exert such a great force at the other. Now physicists all agree that Newton's Three Laws are the basis from which all of mechanics follows, but if you ask for an explanation of the lever in terms of these three laws, you will almost certainly not get a satisfactory answer.

Consider now the Milky Way. Here also we see an innumerable dust, only the grains of this dust are no longer atoms but stars; these grains also move with great velocities, they act at a distance one upon another, but this action is so slight at great distances that their trajectories are rectilineal; nevertheless, from time to time, two of them may come near enough together to be deviated from their course, like a comet that passed too close to Jupiter. In a word, in the eyes of a giant, to whom our Suns were what our atoms are to us, the Milky Way would only look like a bubble of gas.

It is the heart which experiences God and not the reason.

In any case, do you really think kids even want something that is relevant to their daily lives? You think something practical like compound interest is going to get them excited? People enjoy fantasy, and that is just what mathematics can provide -- a relief from daily life, an anodyne to the practical workaday world.

The unity of all science consists alone in its method, not in its material.

Would you tell me, please, which way I ought to go from here?""That depends a good deal on where you want to get to.""I don't much care where –""Then it doesn't matter which way you go.

A great many worries can be diminished by realizing the unimportance of the matter which is causing anxiety.

I do not feel like an alien in this universe. The more I examine the universe and study the details of its architecture, the more evidence I find that the universe is some sense must have known that we were coming

Do you wish men to speak well of you? Then never speak well of yourself.

The Moving Finger writes; and, having writ,Moves on: nor all thy Piety nor WitShall lure it back to cancel half a Line,Nor all thy Tears wash out a Word of it.

The full impact of the Lobachevskian method of challenging axioms has probably yet to be felt. It is no exaggeration to call Lobachevsky the Copernicus of Geometry [as did Clifford], for geometry is only a part of the vaster domain which he renovated; it might even be just to designate him as a Copernicus of all thought.

Like a Shakespearean sonnet that captures the very essence of love, or a painting that brings out the beauty of the human form that is far more than just skin deep, Euler’s equation reaches down into the very depths of existence.

Whenever explaining an event, we must choose from three competing modes of explanation. These are regularity, chance, and design... To attribute an event to design is to say that it cannot reasonably be referred to either regularity or chance.

A Puritan twist in our nature makes us think that anything good for us must be twice as good if it's hard to swallow. Learning Greek and Latin used to play the role of character builder, since they were considered to be as exhausting and unrewarding as digging a trench in the morning and filling it up in the afternoon. It was what made a man, or a woman -- or more likely a robot -- of you. Now math serves that purpose in many schools: your task is to try to follow rules that make sense, perhaps, to some higher beings; and in the end to accept your failure with humbled pride. As you limp off with your aching mind and bruised soul, you know that nothing in later life will ever be as difficult.What a perverse fate for one of our kind's greatest triumphs! Think how absurd it would be were music treated this way (for math and music are both excursions into sensuous structure): suffer through playing your scales, and when you're an adult you'll never have to listen to music again. And this is mathematics we're talking about, the language in which, Galileo said, the Book of the World is written. This is mathematics, which reaches down into our deepest intuitions and outward toward the nature of the universe -- mathematics, which explains the atoms as well as the stars in their courses, and lets us see into the ways that rivers and arteries branch. For mathematics itself is the study of connections: how things ideally must and, in fact, do sort together -- beyond, around, and within us. It doesn't just help us to balance our checkbooks; it leads us to see the balances hidden in the tumble of events, and the shapes of those quiet symmetries behind the random clatter of things. At the same time, we come to savor it, like music, wholly for itself. Applied or pure, mathematics gives whoever enjoys it a matchless self-confidence, along with a sense of partaking in truths that follow neither from persuasion nor faith but stand foursquare on their own. This is why it appeals to what we will come back to again and again: our **architectural instinct** -- as deep in us as any of our urges.

Ability and necessity dwell near each other.

Remember your humanity, and forget the rest.

Few men speak humbly of humility, chastely of chastity, skeptically of skepticism.

Religion divides us, while it is our human characteristics that bind us to each other.

A BOAT beneath a sunny sky,Lingering onward dreamilyIn an evening of July —Children three that nestle near,Eager eye and willing ear,Pleased a simple tale to hear —Long has paled that sunny sky:Echoes fade and memories die:Autumn frosts have slain July.Still she haunts me, phantomwise,Alice moving under skiesNever seen by waking eyes.Children yet, the tale to hear,Eager eye and willing ear,Lovingly shall nestle near.In a Wonderland they lie,Dreaming as the days go by,Dreaming as the summers die:Ever drifting down the stream —Lingering in the golden gleam —Life, what is it but a dream?

The probability of an event is the reason we have to believe that it has taken place, or that it will take place.The measure of the probability of an event is the ratio of the number of cases favourable to that event, to the total number of cases favourable or contrary, and all equally possible.

Aren't you sometimes frightened at being planted out here, with nobody to take care of you?''There's the tree in the middle,' said the Rose:'what else is it good for?''But what could it do, if any danger came?' Alice asked.'It could bark,' said the Rose.

Humans are actually spirit beings who each possess a mind and a body and what we do with them is entirely up to us.

It's only in the mysterious equations of love that any logical reasons can be found.

Those who fear life are already three parts dead.

O friend, for the morrow let us not worryThis moment we have now, let us not hurryWhen our time comes, we shall not tarryWith seven thousand-year-olds, our burden carry

Neither a man nor a crowd nor a nation can be trusted to act humanely or to think sanely under the influence of a great fear.

How, in such an alien and inhuman world, can so powerless a creature as man preserve his aspirations untarnished? A strange mystery it is that nature, omnipotent but blind, in the revolutions of her secular hurryings through the abysses of space, has brought forth at last a child, subject still to her power, but gifted with sight, with knowledge of good and evil, with the capacity of judging all the works of his unthinking mother. In spite of death, the mark and seal of the parental control, man is yet free, during his brief years, to examine, to criticize, to know, and in imagination to create. To him alone, in the world with which he is aquainted, this freedom belongs; and in this lies his superiority to the resistless forces that control his outward life.

Give me a lever long enough And a prop strong enough I can single-handed move the world.

Grammar and ordinary language are bad guides to metaphysics. A great book might be written showing the influence of syntax on philosophy.

The simple process of eating and breathing weave all of us together into a vast four-dimensional array. No matter how isolated you may sometimes feel, no matter how lonely, you are never really cut off from the whole.

The twelve jurors were all writing very busily on the slates. "What are they doing?" Alice whispered to the Gryphon. "They can't have anything to put down yet, before the trial's begun.""They're putting down their names," the Gryphon whispered in reply, "for fear they should forget them before the end of the trial.

No mathematician in the world would bother making these senseless distinctions: 2 1/2 is a "mixed number " while 5/2 is an "improper fraction." They're EQUAL for crying out loud. They are the exact same numbers and have the exact same properties. Who uses such words outside of fourth grade?

Travelling, whether in the mental or the physical world, is a joy, and it is good to know that, in the mental world at least, there are vast countries still very imperfectly explored

But without going to such extremes prudence may easily involve the loss of some of the best things in life. The worshipper of Dionysus reacts against prudence. In intoxication, physical or spiritual, he recovers an intensity of feeling which prudence had destroyed; he finds the world full of delight and beauty, and his imagination is suddenly liberated from the prison of every-day preoccupations

As we are always preparing to be happy it is inevitable that we should never be so.